Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) order 3072 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C3 ⋊ #557 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D3 |
| commutator, socle, prime radical, cosocle | C22×C3 | C24 |
| Fitting | C3×#125 | C2 |
| hypercenter | #125 | D3 |
| hypocenter | C3 | #557 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ #125 | |
200 conjugacy classes of subgroups; only the 105 normal subgroups are shown.
This group has 200 conjugacy classes of subgroups — too many to draw. Only its 105 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 19 points:
(1,2,3)(4,7,8,11)(5,9,6,10)(12,15)(13,17)(14,18)(16,19);(4,6,8,5)(7,9,11,10)(12,14)(13,15)(16,17)(18,19);(12,14,16,17)(13,15,18,19);(2,3)(4,5,8,6)(7,10,11,9)(12,13,16,18)(14,15,17,19)Polycyclic presentation:
2^6,3|a^2=f,b^2=e,c^2=ef,d^2=ef,d^a=def,g^a=g^2,c^b=ce,d^c=df39 conjugacy classes — view the full character table.
| maximal subgroup of | #81151, #81153, #81155, … |
| maximal normal subgroup of | #81151, #81153, #81155, … |
| maximal quotient of | #74157, #74158, #74169, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 1244);